Problem: What is the area of the region between the graphs of $f(x)=\dfrac{4}{x}$ and $g(x)=5$ from $x=-6$ to $x=-2$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $6$ (Choice B) B $20$ (Choice C) C $24\ln(6)-10$ (Choice D) D $20+\ln(81)$
Answer: Visualizing the area We sketch the graphs of $f$ and $g$ first. ${\llap{-}2}$ ${\llap{-}4}$ ${\llap{-}6}$ ${2}$ ${4}$ ${\llap{-}2}$ ${\llap{-}4}$ $f$ $g$ $y$ $x$ From the graph, it appears that $g(x)\ge f(x)$ between $x=-6$ and $x=-2$. From this we are looking to evaluate: $ \int_{-6}^{-2}\left( g(x)-f(x) \right)\,dx$ Evaluating the definite integral $\begin{aligned} &\phantom{=} \int_{-6}^{-2} \left( 5- \left(\dfrac{4}{x} \right) \right) \,dx \\\\ &= 5x - 4\ln|x| ~\Bigg|_{-6}^{-2} \\\\ &= \left( -10 -4\ln(2) \right) -\left( -30-4\ln(6) \right)\\\\ &=20-\ln(81) \end{aligned}$ [Where did the 81 come from?] Answer The area is $20+\ln(81)$ square units.